Prove identity then evaluate integral

A question is this type if and only if it has multiple parts where the first part proves a trigonometric identity, and subsequent parts use that identity to evaluate definite integrals.

3 questions · Standard +0.3

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CAIE P3 2010 June Q4
6 marks Standard +0.3
4
  1. Using the expansions of \(\cos ( 3 x - x )\) and \(\cos ( 3 x + x )\), prove that $$\frac { 1 } { 2 } ( \cos 2 x - \cos 4 x ) \equiv \sin 3 x \sin x$$
  2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin 3 x \sin x \mathrm {~d} x = \frac { 1 } { 8 } \sqrt { } 3$$
CAIE P3 2010 June Q7
8 marks Standard +0.3
7
  1. Prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
  2. Using this result, find the exact value of $$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } \theta \mathrm {~d} \theta$$
CAIE P3 2021 November Q6
6 marks Standard +0.3
6
  1. Using the expansions of \(\sin ( 3 x + 2 x )\) and \(\sin ( 3 x - 2 x )\), show that $$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )\).